Large-Maturity Regimes of the Heston Forward Smile

TL;DR

This paper characterizes the behavior of the implied volatility smile for large maturities in the Heston model, revealing complex parameter-dependent asymptotics and extending existing volatility approximation formulas.

Contribution

It provides a comprehensive analysis of large-maturity forward implied volatility in the Heston model, including new asymptotics and extended SVI-type formulas, with refined large deviations techniques.

Findings

Derived new implied volatility asymptotics for large maturities.

Identified parameter-dependent algebraic expansion terms.

Extended SVI-type formulas for forward and spot cases.

Abstract

We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and different powers of the maturity come into play. As a by-product of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVI-type formulae. The proofs are based on extensions and refinements of sharp large deviations theory, in particular in cases where standard convexity arguments fail.